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52-397: A regular hexagon has Schläfli symbol {6} and can also be constructed as a truncated equilateral triangle , t{3}, which alternates two types of edges. A regular hexagon is defined as a hexagon that is both equilateral and equiangular . It is bicentric , meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). The common length of

104-514: A = 1, this produces the following table: ( Since cot ⁡ x → 1 / x {\displaystyle \cot x\rightarrow 1/x} as x → 0 {\displaystyle x\rightarrow 0} , the area when s = 1 {\displaystyle s=1} tends to n 2 / 4 π {\displaystyle n^{2}/4\pi } as n {\displaystyle n} grows large.) Of all n -gons with

156-409: A hexagram . A regular hexagon can be dissected into six equilateral triangles by adding a center point. This pattern repeats within the regular triangular tiling . A regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. This pattern repeats within the rhombitrihexagonal tiling . There are six self-crossing hexagons with

208-429: A circle is said to circumscribe the points or a polygon formed from them; such a polygon is said to be inscribed in the circle. Circumcircle , the circumscribed circle of a triangle, which always exists for a given triangle. Cyclic polygon , a general polygon that can be circumscribed by a circle. The vertices of this polygon are concyclic points . All triangles are cyclic polygons. Cyclic quadrilateral ,

260-435: A circle. The value of the internal angle can never become exactly equal to 180°, as the circumference would effectively become a straight line (see apeirogon ). For this reason, a circle is not a polygon with an infinite number of sides. For n > 2, the number of diagonals is 1 2 n ( n − 3 ) {\displaystyle {\tfrac {1}{2}}n(n-3)} ; i.e., 0, 2, 5, 9, ..., for

312-416: A given perimeter, the one with the largest area is regular. Some regular polygons are easy to construct with compass and straightedge ; other regular polygons are not constructible at all. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon. This led to

364-421: A hexagon has vertices on the circumcircle of an acute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle. Let ABCDEF be a hexagon formed by six tangent lines of a conic section. Then Brianchon's theorem states that the three main diagonals AD, BE, and CF intersect at

416-434: A regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then PE + PF = PA + PB + PC + PD . It follows from the ratio of circumradius to inradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long diagonal of 1.0000000 will have a distance of 0.8660254 or cos(30°) between parallel sides. For an arbitrary point in

468-574: A regular hexagonal pattern. The two simple roots have a 120° angle between them. The 12 roots of the Exceptional Lie group G2 , represented by a Dynkin diagram [REDACTED] [REDACTED] [REDACTED] are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them. Coxeter states that every zonogon (a 2 m -gon whose opposite sides are parallel and of equal length) can be dissected into 1 ⁄ 2 m ( m − 1) parallelograms. In particular this

520-420: A single point. In a hexagon that is tangential to a circle and that has consecutive sides a , b , c , d , e , and f , If an equilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the centroids of opposite triangles form another equilateral triangle. A skew hexagon is a skew polygon with six vertices and edges but not existing on

572-503: A special case of a cyclic polygon. See also [ edit ] Smallest-circle problem , the related problem of finding the circle with minimal radius containing an arbitrary set of points, not necessarily passing through them. Inscribed figure [REDACTED] Index of articles associated with the same name This set index article includes a list of related items that share the same name (or similar names). If an internal link incorrectly led you here, you may wish to change

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624-447: A triangle, square, pentagon, hexagon, ... . The diagonals divide the polygon into 1, 4, 11, 24, ... pieces OEIS :  A007678 . For a regular n -gon inscribed in a unit-radius circle, the product of the distances from a given vertex to all other vertices (including adjacent vertices and vertices connected by a diagonal) equals n . For a regular simple n -gon with circumradius R and distances d i from an arbitrary point in

676-461: A uniform antiprism . All edges and internal angles are equal. More generally regular skew polygons can be defined in n -space. Examples include the Petrie polygons , polygonal paths of edges that divide a regular polytope into two halves, and seen as a regular polygon in orthogonal projection. In the infinite limit regular skew polygons become skew apeirogons . A non-convex regular polygon

728-731: Is 2 nR − ⁠ 1 / 4 ⁠ ns , where s is the side length and R is the circumradius. If d i {\displaystyle d_{i}} are the distances from the vertices of a regular n {\displaystyle n} -gon to any point on its circumcircle, then Coxeter states that every zonogon (a 2 m -gon whose opposite sides are parallel and of equal length) can be dissected into ( m 2 ) {\displaystyle {\tbinom {m}{2}}} or ⁠ 1 / 2 ⁠ m ( m − 1) parallelograms. These tilings are contained as subsets of vertices, edges and faces in orthogonal projections m -cubes . In particular, this

780-401: Is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex , star or skew . In the limit , a sequence of regular polygons with an increasing number of sides approximates a circle , if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line ), if

832-474: Is a regular star polygon . The most common example is the pentagram , which has the same vertices as a pentagon , but connects alternating vertices. For an n -sided star polygon, the Schläfli symbol is modified to indicate the density or "starriness" m of the polygon, as { n / m }. If m is 2, for example, then every second point is joined. If m is 3, then every third point is joined. The boundary of

884-465: Is a uniform polyhedron which has just one kind of face. The remaining (non-uniform) convex polyhedra with regular faces are known as the Johnson solids . A polyhedron having regular triangles as faces is called a deltahedron . Circumscribed circle In geometry, a circumscribed circle for a set of points is a circle passing through each of them. Such

936-750: Is full symmetry, and a1 is no symmetry. p6 , an isogonal hexagon constructed by three mirrors can alternate long and short edges, and d6 , an isotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are duals of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an elongated rhombus , while d2 and p2 can be seen as horizontally and vertically elongated kites . g2 hexagons, with opposite sides parallel are also called hexagonal parallelogons . Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only

988-439: Is inscribed in any conic section , and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration. The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point . If

1040-447: Is no Platonic solid made of only regular hexagons, because the hexagons tessellate , not allowing the result to "fold up". The Archimedean solids with some hexagonal faces are the truncated tetrahedron , truncated octahedron , truncated icosahedron (of soccer ball and fullerene fame), truncated cuboctahedron and the truncated icosidodecahedron . These hexagons can be considered truncated triangles, with Coxeter diagrams of

1092-442: Is the Petrie polygon for these higher dimensional regular , uniform and dual polyhedra and polytopes, shown in these skew orthogonal projections : A principal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex equilateral hexagon (one with all sides equal) with common side a , there exists a principal diagonal d 1 such that and a principal diagonal d 2 such that There

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1144-425: Is the distance from an arbitrary point in the plane to the centroid of a regular n {\displaystyle n} -gon with circumradius R {\displaystyle R} , then where m {\displaystyle m} = 1, 2, …, n − 1 {\displaystyle n-1} . For a regular n -gon, the sum of the perpendicular distances from any interior point to

1196-406: Is true for regular polygons with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a Petrie polygon projection of a cube , with 3 of 6 square faces. Other parallelogons and projective directions of the cube are dissected within rectangular cuboids . A regular hexagon has Schläfli symbol {6}. A regular hexagon is a part of

1248-412: Is true for any regular polygon with an even number of sides, in which case the parallelograms are all rhombi. The list OEIS :  A006245 gives the number of solutions for smaller polygons. The area A of a convex regular n -sided polygon having side s , circumradius R , apothem a , and perimeter p is given by For regular polygons with side s = 1, circumradius R = 1, or apothem

1300-460: The Gauss–Wantzel theorem . Equivalently, a regular n -gon is constructible if and only if the cosine of its common angle is a constructible number —that is, can be written in terms of the four basic arithmetic operations and the extraction of square roots. A regular skew polygon in 3-space can be seen as nonplanar paths zig-zagging between two parallel planes, defined as the side-edges of

1352-484: The dihedral group D 6 . The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a triangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon is equilateral , and that the regular hexagon can be partitioned into six equilateral triangles. Like squares and equilateral triangles , regular hexagons fit together without any gaps to tile

1404-462: The g6 subgroup has no degrees of freedom but can be seen as directed edges . Hexagons of symmetry g2 , i4 , and r12 , as parallelogons can tessellate the Euclidean plane by translation. Other hexagon shapes can tile the plane with different orientations. The 6 roots of the simple Lie group A2 , represented by a Dynkin diagram [REDACTED] [REDACTED] [REDACTED] , are in

1456-476: The n sides is n times the apothem (the apothem being the distance from the center to any side). This is a generalization of Viviani's theorem for the n = 3 case. The circumradius R from the center of a regular polygon to one of the vertices is related to the side length s or to the apothem a by For constructible polygons , algebraic expressions for these relationships exist (see Bicentric polygon § Regular polygons ) . The sum of

1508-542: The vertex arrangement of the regular hexagon: From bees' honeycombs to the Giant's Causeway , hexagonal patterns are prevalent in nature due to their efficiency. In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest hexagons. This means that honeycombs require less wax to construct and gain much strength under compression . Irregular hexagons with parallel opposite edges are called parallelogons and can also tile

1560-494: The area can also be expressed in terms of the apothem a and the perimeter p . For the regular hexagon these are given by a = r , and p = 6 R = 4 r 3 {\displaystyle {}=6R=4r{\sqrt {3}}} , so The regular hexagon fills the fraction 3 3 2 π ≈ 0.8270 {\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270} of its circumscribed circle . If

1612-401: The constructibility of regular polygons: (A Fermat prime is a prime number of the form 2 ( 2 n ) + 1. {\displaystyle 2^{\left(2^{n}\right)}+1.} ) Gauss stated without proof that this condition was also necessary , but never published his proof. A full proof of necessity was given by Pierre Wantzel in 1837. The result is known as

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1664-433: The edge length is fixed. These properties apply to all regular polygons, whether convex or star : The symmetry group of an n -sided regular polygon is dihedral group D n (of order 2 n ): D 2 , D 3 , D 4 , ... It consists of the rotations in C n , together with reflection symmetry in n axes that pass through the center. If n is even then half of these axes pass through two opposite vertices, and

1716-546: The etymology of the term. The prefix "hex-" originates from the Greek word "hex," meaning six, while "sex-" comes from the Latin "sex," also signifying six. Some linguists and mathematicians argue that since many English mathematical terms derive from Latin, the use of "sexagon" would align with this tradition. Historical discussions date back to the 19th century, when mathematicians began to standardize terminology in geometry. However,

1768-440: The form [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] and [REDACTED] [REDACTED] [REDACTED] [REDACTED] [REDACTED] . There are other symmetry polyhedra with stretched or flattened hexagons, like these Goldberg polyhedron G(2,0): There are also 9 Johnson solids with regular hexagons: The debate over whether hexagons should be referred to as "sexagons" has its roots in

1820-445: The hexagon), D , is twice the maximal radius or circumradius , R , which equals the side length, t . The minimal diameter or the diameter of the inscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), d , is twice the minimal radius or inradius , r . The maxima and minima are related by the same factor: The area of a regular hexagon For any regular polygon ,

1872-466: The other half through the midpoint of opposite sides. If n is odd then all axes pass through a vertex and the midpoint of the opposite side. All regular simple polygons (a simple polygon is one that does not intersect itself anywhere) are convex. Those having the same number of sides are also similar . An n -sided convex regular polygon is denoted by its Schläfli symbol { n }. For n < 3, we have two degenerate cases: In certain contexts all

1924-404: The perpendiculars from a regular n -gon's vertices to any line tangent to the circumcircle equals n times the circumradius. The sum of the squared distances from the vertices of a regular n -gon to any point on its circumcircle equals 2 nR where R is the circumradius. The sum of the squared distances from the midpoints of the sides of a regular n -gon to any point on the circumcircle

1976-413: The plane (three hexagons meeting at every vertex), and so are useful for constructing tessellations . The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. The maximal diameter (which corresponds to the long diagonal of

2028-506: The plane by translation. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3-space by translation. In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane. Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon

2080-402: The plane of a regular hexagon with circumradius R {\displaystyle R} , whose distances to the centroid of the regular hexagon and its six vertices are L {\displaystyle L} and d i {\displaystyle d_{i}} respectively, we have If d i {\displaystyle d_{i}} are the distances from

2132-441: The plane to the vertices, we have For higher powers of distances d i {\displaystyle d_{i}} from an arbitrary point in the plane to the vertices of a regular n {\displaystyle n} -gon, if then and where m {\displaystyle m} is a positive integer less than n {\displaystyle n} . If L {\displaystyle L}

Hexagon - Misplaced Pages Continue

2184-573: The polygon winds around the center m times. The (non-degenerate) regular stars of up to 12 sides are: m and n must be coprime , or the figure will degenerate. The degenerate regular stars of up to 12 sides are: Depending on the precise derivation of the Schläfli symbol, opinions differ as to the nature of the degenerate figure. For example, {6/2} may be treated in either of two ways: All regular polygons are self-dual to congruency, and for odd n they are self-dual to identity. In addition,

2236-506: The polygons considered will be regular. In such circumstances it is customary to drop the prefix regular. For instance, all the faces of uniform polyhedra must be regular and the faces will be described simply as triangle, square, pentagon, etc. For a regular convex n -gon, each interior angle has a measure of: and each exterior angle (i.e., supplementary to the interior angle) has a measure of 360 n {\displaystyle {\tfrac {360}{n}}} degrees, with

2288-423: The question being posed: is it possible to construct all regular n -gons with compass and straightedge? If not, which n -gons are constructible and which are not? Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae . This theory allowed him to formulate a sufficient condition for

2340-517: The regular hexagonal tiling , {6,3}, with three hexagonal faces around each vertex. A regular hexagon can also be created as a truncated equilateral triangle , with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D 3 symmetry. A truncated hexagon, t{6}, is a dodecagon , {12}, alternating two types (colors) of edges. An alternated hexagon, h{6}, is an equilateral triangle , {3}. A regular hexagon can be stellated with equilateral triangles on its edges, creating

2392-417: The regular star figures (compounds), being composed of regular polygons, are also self-dual. A uniform polyhedron has regular polygons as faces, such that for every two vertices there is an isometry mapping one into the other (just as there is for a regular polygon). A quasiregular polyhedron is a uniform polyhedron which has just two kinds of face alternating around each vertex. A regular polyhedron

2444-539: The same plane. The interior of such a hexagon is not generally defined. A skew zig-zag hexagon has vertices alternating between two parallel planes. A regular skew hexagon is vertex-transitive with equal edge lengths. In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of a triangular antiprism with the same D 3d , [2,6] symmetry, order 12. The cube and octahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons. The regular skew hexagon

2496-431: The sides equals the radius of the circumscribed circle or circumcircle , which equals 2 3 {\displaystyle {\tfrac {2}{\sqrt {3}}}} times the apothem (radius of the inscribed circle ). All internal angles are 120 degrees . A regular hexagon has six rotational symmetries ( rotational symmetry of order six ) and six reflection symmetries ( six lines of symmetry ), making up

2548-399: The successive sides of a cyclic hexagon are a , b , c , d , e , f , then the three main diagonals intersect in a single point if and only if ace = bdf . If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles are concurrent . If

2600-420: The sum of the exterior angles equal to 360 degrees or 2π radians or one full turn. As n approaches infinity, the internal angle approaches 180 degrees. For a regular polygon with 10,000 sides (a myriagon ) the internal angle is 179.964°. As the number of sides increases, the internal angle can come very close to 180°, and the shape of the polygon approaches that of a circle. However the polygon can never become

2652-412: The term "hexagon" has prevailed in common usage and academic literature, solidifying its place in mathematical terminology despite the historical argument for "sexagon." The consensus remains that "hexagon" is the appropriate term, reflecting its Greek origins and established usage in mathematics. (see Numeral_prefix#Occurrences ). Regular polygon In Euclidean geometry , a regular polygon

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2704-420: The vertices of a regular hexagon to any point on its circumcircle, then The regular hexagon has D 6 symmetry. There are 16 subgroups. There are 8 up to isomorphism: itself (D 6 ), 2 dihedral: (D 3, D 2 ), 4 cyclic : (Z 6 , Z 3 , Z 2 , Z 1 ) and the trivial (e) These symmetries express nine distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order. r12

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